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Origins of Star Wars
Jul. 16th, 2004 10:56 amAfter find the site, I don't want to lose it again, so I'm posting it here for any to read! (and for me to find later)
Origins of Star Wars
Origins of Star Wars
Jul. 16th, 2004
A very small universe
Jul. 16th, 2004 11:59 pmA very small universe (but not the smallest)
In some sense, mathematics consists of objects, their properties (including relationships), and transformations (functions).
In trying to conceive of a very abstract (i.e., very few defined properties) simplified mathematics, consider a universe with one object. It's only property is that it is itself, and the only possible function from that universe to that same universe is the mapping from the lone object to itself. Pretty boring.
Now consider another universe, containing two objects, which I will call "A" and "B." We only want necessary properties, so that we are as abstract as possible, and we want to ensure that we really have two objects and not two names for the same object. How's this: A has the property of being A, and B has the property of not being A. (Saying A is A, and B is B is insufficient, for it doesn't exclude A's being B. Similar for saying A is not B and B is not A.) Then there are four possible functions from this universe into itself: A to A and B to A; A to A and B to B; A to B and B to B; and A to B and B to A.
Actually, in this last example, there is only one property, that of being A. B clearly has no A-ness, while A does. Which leads to the question, "How does B poop?"
In some sense, mathematics consists of objects, their properties (including relationships), and transformations (functions).
In trying to conceive of a very abstract (i.e., very few defined properties) simplified mathematics, consider a universe with one object. It's only property is that it is itself, and the only possible function from that universe to that same universe is the mapping from the lone object to itself. Pretty boring.
Now consider another universe, containing two objects, which I will call "A" and "B." We only want necessary properties, so that we are as abstract as possible, and we want to ensure that we really have two objects and not two names for the same object. How's this: A has the property of being A, and B has the property of not being A. (Saying A is A, and B is B is insufficient, for it doesn't exclude A's being B. Similar for saying A is not B and B is not A.) Then there are four possible functions from this universe into itself: A to A and B to A; A to A and B to B; A to B and B to B; and A to B and B to A.
Actually, in this last example, there is only one property, that of being A. B clearly has no A-ness, while A does. Which leads to the question, "How does B poop?"